Sturm Liouville Problem with Moving Discontinuity Points

In this paper, we present a new discontinuous Sturm Liouville problem with symmetrically located discontinuities which are defined depending on a neighborhood of a midpoint of the interval. Also the problem contains an eigenparameter in one of the boundary conditions and has coupled transmission conditions at the discontinuity points. We investigate the properties of the eigenvalues, obtain asymptotic formulas for the eigenvalues and the corresponding eigenfunctions and construct Green's function of this problem.


Introduction
The Sturm-Liouville theory plays an important role in solving many mathematical physics problems [1]. Such research is motivated by theory of heat and mass transfer, vibrating string problems when the string is loaded additionally with point masses. Also, some problems with transmission conditions arise in thermal conduction problems for a thin laminated plate and inhomogenous materials ( see [2,3]; see also wherein references). Heat conduction problems in composite walls have been analysed by several researcher [4][5][6]. In all of these works, temperature distribution of composite walls involving two or more layers in are investigated spectrally and the results are formulated by the eigenvalues and eigenfunctions of the auxiliary spectral problem. In [7], the presence of infinite number of eigenvalues has been shown and an asymptotic formula has been obtained for the eigenvalues of the spectral problem for the temperature distribution in composite walls. The problem in this work included a twolayer composite wall consisting of different materials, having a common contact surface. When using Fourier's method for heat transfer problems, it becomes necessary to solve a related Sturm Liouville problem in which the separation constant plays the role of a spectral parameter (or eigenparameter). In the case of inhomogeneous materials, the Sturm Liouville equation has variable and not necessarily continuous coefficients so that transmission conditions across the interfaces should be added in the problem. In the present work, we consider a composite medium consisting of two interfaces that are located symmetrically, thus we add to the problem two coupled transmission conditions.
Throughout this paper we consider the boundary value problem with an eigenparameter dependent on one of the boundary conditions and coupled transmission conditions at the points of discontinuity that are    and    , where ;  is a spectral parameter;   q x is a given real valued function which is continuous in , , , , , , 1, 2 are real numbers such that 1 Also for convenience we will use the notations We introduce a new discontinuous Sturm Liouville problem with symmetrically located discontinuities which are defined depending on a neighborhood of a midpoint of the interval.  is a parameter controling the change of neighborhood process (it can be called tuning parameter) and by using the change of this  parameter it's possible to determine points of discontinuity. That is, two points of discontinuity can be determined  [11,12] these problems do not contain an eigenparameter in the boundary conditions. We extend some classic results of Sturm-Liouville theory to the new symmetric and moving discontinuous case. So that, we define a linear operator A in a suitable Hilbert space H such that the eigenvalues of the problem (1)-(7) coincide with those of A , construct a special fundamental system of solutions, obtain the asymptotic formulas for the eigenvalues and the corresponding eigenfunctions that depend on  parameter. Finally, we construct Green's function for the problem (1)-(7).

An Operator Formulation
In this section we will introduce the special inner product in the Hilbert space   2 , L a b   and a symmetric linear operator A defined on this Hilbert space such a way that the problem (1)-(7) can be considered as the eigenvalue problem of this operator.

Definition 1. We define a Hilbert space H of two component vectors by
, : For : , , , and ,

Definition 2. We define a linear operator
Now we can rewrite the problem (1)- (7) in the operator form as AF The eigenvalues and eigenfunctions of the problem (1)- (7) are defined as the eigenvalues and the first components of the corresponding eigenelements of the operator A , respectively.

Lemma 3. The operator A in H is symmetric.
By two partial integration we obtain where, as usual, by   , ; W f g x we denote the Wronskian of the functions f and g :  .
Since f and g satisfy the boundary condition (2), it follows that from the transmission conditions (4)- (7) we get Further, it is easy to verify that Finally, substituting (16)- (19) in (15) We will define two solutions After defining this solution, we may define the solution Hence, , the boundary condition (2) and the transmission conditions (4)- (7).
Analogically, first we define the solution Again, after defining this solution, we define the solution After defining this solution, we define the solution Hence, , the boundary condition (3) and the transmission conditions (4)-(7).

Let us consider the Wronskians
which are independent of    , , where at least one of the constants   (25) into account, it follows that the determinant of this system is

Asymptotic Approximate Formulas
It is known [21] that   x   be the solution of the equation (1) can be written in the form and for 0,1 k  .
We also use Titchmarsh's formulas [19] for asymptotic behaviour of to get the following formulas for the problem (1)-(7) as    : then     has the following asymptotic representations:

Corollary 7. The eigenvalues of the problem (1)-(7) is bounded from below.
We are now ready to find the asymptotic approximation formulas for the eigenvalues of the problem (1)- (7). Since the eigenvalues coincide with the zeros of the entire functions     , it follows that they have no finite accumulation point. Morever, all eigenvalues are real and bounded below by Corollaries 4 and 7. Therefore, we may renumber them as 0    Then from (32)-(37) (for 0 k  ) and the above theorem, the asymptotic behaviour of the eigenfunctions of the problem (1)-(7) is given by: x a O x a a n  x